Bregman iteration ($u^{k+1}\in \arg {\min }_u\{\frac{1}{2}\Vert Ku-f{\Vert }^2+\alpha D_J^{p^k}(u,u^k)\}$)

Starting with $u^0=0$ and $p^0=0\in \partial J(0)$, update $p^{k+1}=p^k+K^{\ast }(f-K{u}^{k+1})/\alpha$. The optimality condition makes $p^{k+1}\in \partial J(u^{k+1})$. The iteration replaces a fixed penalty by its Bregman distance from the preceding iterate and accumulates residual information in the dual variable. The first update uses $k=0$.

For a generalized singular vector, exact data $f=\gamma K{u}_{\lambda }$, and fixed $\alpha >0$, a compatible branch of Bregman iteration has $p^k=\min \{k\gamma /\alpha ,\lambda \}{K}^{\ast }K{u}_{\lambda }$ and the displayed $u^k$. Verify these formulas with the subgradient characterization of an absolutely one-homogeneous functional; before activation the dual coefficient remains at most $\lambda$, and after activation it is $\lambda$. The branch reaches $\gamma u_{\lambda }$ at $k_{\ast }=\lceil \alpha \lambda /\gamma \rceil +1$. Every minimizer branch has the same predicted data, since the quadratic data fidelity is strictly convex in $Ku$. Exact recovery of this particular vector for arbitrary branch choices needs uniqueness, for example injectivity of $K$. For $K(x,y)=x$ and $J(x,y)=|x|$, the second coordinate of every minimizer is arbitrary.